If A={1,2,3,4},B={3,4,5,6},
C={5,6,7,8} and D= {7,8,9,10}.
Find (i) `AuuB`
(ii) `AuuC`
(iii) `BuuC`
(iv) `BuuD`
(v) `AuuBuuC`
(vi) `AuuBuuD`
(vii) `BuuCuuD`.

Answer

To solve the given question, we need to find the union of the specified sets. The union of two sets combines all the unique elements from both sets.

Given:
- \( A = \{1, 2, 3, 4\} \)
- \( B = \{3, 4, 5, 6\} \)
- \( C = \{5, 6, 7, 8\} \)
- \( D = \{7, 8, 9, 10\} \)

Now, let's find the unions step by step.

### (i) \( A \cup B \)

To find \( A \cup B \):
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( B \): \( 3, 4, 5, 6 \)

Combining these, we get:
- Unique elements: \( 1, 2, 3, 4, 5, 6 \)

So, \( A \cup B = \{1, 2, 3, 4, 5, 6\} \).

### (ii) \( A \cup C \)

To find \( A \cup C \):
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( C \): \( 5, 6, 7, 8 \)

Combining these, we get:
- Unique elements: \( 1, 2, 3, 4, 5, 6, 7, 8 \)

So, \( A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8\} \).

### (iii) \( B \cup C \)

To find \( B \cup C \):
- Elements in \( B \): \( 3, 4, 5, 6 \)
- Elements in \( C \): \( 5, 6, 7, 8 \)

Combining these, we get:
- Unique elements: \( 3, 4, 5, 6, 7, 8 \)

So, \( B \cup C = \{3, 4, 5, 6, 7, 8\} \).

### (iv) \( B \cup D \)

To find \( B \cup D \):
- Elements in \( B \): \( 3, 4, 5, 6 \)
- Elements in \( D \): \( 7, 8, 9, 10 \)

Combining these, we get:
- Unique elements: \( 3, 4, 5, 6, 7, 8, 9, 10 \)

So, \( B \cup D = \{3, 4, 5, 6, 7, 8, 9, 10\} \).

### (v) \( A \cup B \cup C \)

To find \( A \cup B \cup C \):
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( B \): \( 3, 4, 5, 6 \)
- Elements in \( C \): \( 5, 6, 7, 8 \)

Combining these, we get:
- Unique elements: \( 1, 2, 3, 4, 5, 6, 7, 8 \)

So, \( A \cup B \cup C = \{1, 2, 3, 4, 5, 6, 7, 8\} \).

### (vi) \( A \cup B \cup D \)

To find \( A \cup B \cup D \):
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( B \): \( 3, 4, 5, 6 \)
- Elements in \( D \): \( 7, 8, 9, 10 \)

Combining these, we get:
- Unique elements: \( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \)

So, \( A \cup B \cup D = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).

### (vii) \( B \cup C \cup D \)

To find \( B \cup C \cup D \):
- Elements in \( B \): \( 3, 4, 5, 6 \)
- Elements in \( C \): \( 5, 6, 7, 8 \)
- Elements in \( D \): \( 7, 8, 9, 10 \)

Combining these, we get:
- Unique elements: \( 3, 4, 5, 6, 7, 8, 9, 10 \)

So, \( B \cup C \cup D = \{3, 4, 5, 6, 7, 8, 9, 10\} \).

### Summary of Results:
1. \( A \cup B = \{1, 2, 3, 4, 5, 6\} \)
2. \( A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8\} \)
3. \( B \cup C = \{3, 4, 5, 6, 7, 8\} \)
4. \( B \cup D = \{3, 4, 5, 6, 7, 8, 9, 10\} \)
5. \( A \cup B \cup C = \{1, 2, 3, 4, 5, 6, 7, 8\} \)
6. \( A \cup B \cup D = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
7. \( B \cup C \cup D = \{3, 4, 5, 6, 7, 8, 9, 10\} \)